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Properties

Label 2-4608-16.13-c1-0-51
Degree 22
Conductor 46084608
Sign 0.923−0.382i0.923 - 0.382i
Analytic cond. 36.795036.7950
Root an. cond. 6.065896.06589
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)5-s + 4i·7-s + (4 − 4i)11-s + (3 + 3i)13-s + 6·17-s + (4 + 4i)19-s − 8i·23-s + 3i·25-s + (3 + 3i)29-s − 4·31-s + (4 + 4i)35-s + (1 − i)37-s − 2i·41-s + (4 − 4i)43-s − 8·47-s + ⋯
L(s)  = 1  + (0.447 − 0.447i)5-s + 1.51i·7-s + (1.20 − 1.20i)11-s + (0.832 + 0.832i)13-s + 1.45·17-s + (0.917 + 0.917i)19-s − 1.66i·23-s + 0.600i·25-s + (0.557 + 0.557i)29-s − 0.718·31-s + (0.676 + 0.676i)35-s + (0.164 − 0.164i)37-s − 0.312i·41-s + (0.609 − 0.609i)43-s − 1.16·47-s + ⋯

Functional equation

Λ(s)=(4608s/2 ΓC(s) L(s)=((0.923−0.382i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4608s/2 ΓC(s+1/2) L(s)=((0.923−0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46084608    =    29⋅322^{9} \cdot 3^{2}
Sign: 0.923−0.382i0.923 - 0.382i
Analytic conductor: 36.795036.7950
Root analytic conductor: 6.065896.06589
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4608(1153,⋅)\chi_{4608} (1153, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4608, ( :1/2), 0.923−0.382i)(2,\ 4608,\ (\ :1/2),\ 0.923 - 0.382i)

Particular Values

L(1)L(1) ≈\approx 2.6957906352.695790635
L(12)L(\frac12) ≈\approx 2.6957906352.695790635
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
good5 1+(−1+i)T−5iT2 1 + (-1 + i)T - 5iT^{2}
7 1−4iT−7T2 1 - 4iT - 7T^{2}
11 1+(−4+4i)T−11iT2 1 + (-4 + 4i)T - 11iT^{2}
13 1+(−3−3i)T+13iT2 1 + (-3 - 3i)T + 13iT^{2}
17 1−6T+17T2 1 - 6T + 17T^{2}
19 1+(−4−4i)T+19iT2 1 + (-4 - 4i)T + 19iT^{2}
23 1+8iT−23T2 1 + 8iT - 23T^{2}
29 1+(−3−3i)T+29iT2 1 + (-3 - 3i)T + 29iT^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1+(−1+i)T−37iT2 1 + (-1 + i)T - 37iT^{2}
41 1+2iT−41T2 1 + 2iT - 41T^{2}
43 1+(−4+4i)T−43iT2 1 + (-4 + 4i)T - 43iT^{2}
47 1+8T+47T2 1 + 8T + 47T^{2}
53 1+(7−7i)T−53iT2 1 + (7 - 7i)T - 53iT^{2}
59 1−59iT2 1 - 59iT^{2}
61 1+(3+3i)T+61iT2 1 + (3 + 3i)T + 61iT^{2}
67 1+(8+8i)T+67iT2 1 + (8 + 8i)T + 67iT^{2}
71 1−71T2 1 - 71T^{2}
73 1−10iT−73T2 1 - 10iT - 73T^{2}
79 1−12T+79T2 1 - 12T + 79T^{2}
83 1+(4+4i)T+83iT2 1 + (4 + 4i)T + 83iT^{2}
89 1+16iT−89T2 1 + 16iT - 89T^{2}
97 1−8T+97T2 1 - 8T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.569028267117341858163542905323, −7.84732912415814948894030486778, −6.67591611004168398158490410375, −5.94037993955020654453763340136, −5.71903883616252366042777508005, −4.78481354215264691044535901634, −3.63586968003585771744139013649, −3.08088799824196311276220267706, −1.81042524086582731211086656721, −1.10069760573670121670120322937, 0.955233362683064482232371904822, 1.58221874533219103996538297112, 3.07601062391924803363290300885, 3.64492984390797301479360053883, 4.45528343281635513792574067847, 5.33720277596916778165013425701, 6.23393742597915424383474765373, 6.87265703637919469004909322905, 7.56036711427846394512089693224, 7.947885281363358547833914804977

Graph of the ZZ-function along the critical line